Definition   [1]} A swap matrix  \(W_{[m,n]}\in {\mathcal {M}}_{mn\times mn}\)  is defined as follows:
\(\begin{array}{ccr}W_{[m,n]}&=&\delta _{mn}[1,m+1,\cdots ,(n-1)m+1;\\~&~&2,m+2,\cdots ,(n-1)m+2;\\~&~&\cdots ~;~m,2m,\cdots ,nm].\end{array}\) 

